3 - Coarse graining, entropies and Lyapunov exponents at work

Summary

The meaning of the world is the separation of wish and fact.

Kurt Gödel

In the previous chapter we saw that in deterministic dynamical systems there exist well established ways to define and measure the complexity of a temporal evolution, in terms of either the Lyapunov exponents or the Kolmogorov–Sinai entropy. This approach is rather successful in deterministic low-dimensional systems. On the other hand in high-dimensional systems, as well as in low-dimensional cases without a unique characteristic time (as in the example discussed in Section 2.3.3), some interesting features cannot be captured by the Lyapunov exponents or the Kolmogorov–Sinai entropy. In this chapter we will see how an analysis in terms of the finite size Lyapunov exponents (FSLE) and ∊-entropy, defined in Chapter 2, allows the characterization of non-trivial systems in situations far from asymptotic (i.e. finite time and finite observational resolution). In particular, we will discuss the utility of ∊-entropy and FSLE for a pragmatic classification of signals, and the use of chaotic systems in the generation of sequences of (pseudo) random numbers. In addition we will discuss systems containing some randomness.

Characterization of the complexity and system modeling

Typically in experimental investigations, time records of only few observables are available, and the equations of motion are not known. From a conceptual point of view, this case can be treated in the same framework that is used when the evolution laws are known. Indeed, in principle, with the embedding technique one can reconstruct the topological features of the phase space and dynamics (Takens 1981, Abarbanel et al. 1993, Kantz and Schreiber 1997).